Triangle calculator

Please enter what you know about the triangle:
Symbols definition of ABC triangle

You have entered side a, b and angle α.

Triangle has two solutions: a=3.3; b=5.4; c=2.34395772776 and a=3.3; b=5.4; c=7.80991030269.

#1 Obtuse scalene triangle.

Sides: a = 3.3   b = 5.4   c = 2.34395772776

Area: T = 2.16604929007
Perimeter: p = 11.04395772776
Semiperimeter: s = 5.52197886388

Angle ∠ A = α = 20° = 0.34990658504 rad
Angle ∠ B = β = 145.9677067521° = 145°58'1″ = 2.54876059277 rad
Angle ∠ C = γ = 14.03329324787° = 14°1'59″ = 0.24549208755 rad

Height: ha = 1.30993896368
Height: hb = 0.88001825558
Height: hc = 1.8476908774

Median: ma = 3.82202501121
Median: mb = 0.94443574106
Median: mc = 4.31993280196

Inradius: r = 0.39114086285
Circumradius: R = 4.82442772603

Vertex coordinates: A[2.34395772776; 0] B[0; 0] C[-2.73547628747; 1.8476908774]
Centroid: CG[-0.13217285324; 0.6165636258]
Coordinates of the circumscribed circle: U[1.17697886388; 4.68803040098]
Coordinates of the inscribed circle: I[0.12197886388; 0.39114086285]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 160° = 0.34990658504 rad
∠ B' = β' = 34.03329324787° = 34°1'59″ = 2.54876059277 rad
∠ C' = γ' = 165.9677067521° = 165°58'1″ = 0.24549208755 rad


How did we calculate this triangle?

1. Input data entered: side a, b and angle α.

a = 3.3 ; ; b = 5.4 ; ; alpha = 20° ; ;

2. From angle α, side b and side a we calculate side c - by using the law of cosines and quadratic equation:

a**2 = b**2 + c**2 - 2b c cos alpha ; ; ; ; 3.3**2 = 5.4**2 + c**2 - 2 * 5.4 * c * cos 20° ; ; ; ; ; ; c**2 -10.149c +18.27 =0 ; ; p=1; q=-10.149; r=18.27 ; ; D = q**2 - 4pr = 10.149**2 - 4 * 1 * 18.27 = 29.9157119227 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 10.15 ± sqrt{ 29.92 } }{ 2 } ; ; c_{1,2} = 5.07434015 ± 2.73476287467 ; ; c_{1} = 7.80910302467 ; ; c_{2} = 2.33957727533 ; ; ; ;
 text{ Factored form: } ; ; (c -7.80910302467) (c -2.33957727533) = 0 ; ; ; ; c > 0 ; ; ; ; c = 7.809 ; ;
Now we know the lengths of all three sides of the triangle and the triangle is uniquely determined. Next we calculate another its characteristics - same procedure as calculation of the triangle from the known three sides SSS.

a = 3.3 ; ; b = 5.4 ; ; c = 2.34 ; ;

3. The triangle circumference is the sum of the lengths of its three sides

p = a+b+c = 3.3+5.4+2.34 = 11.04 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 11.04 }{ 2 } = 5.52 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 5.52 * (5.52-3.3)(5.52-5.4)(5.52-2.34) } ; ; T = sqrt{ 4.67 } = 2.16 ; ;

6. Calculate the heights of the triangle from its area.

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 2.16 }{ 3.3 } = 1.31 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 2.16 }{ 5.4 } = 0.8 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 2.16 }{ 2.34 } = 1.85 ; ;

7. Calculation of the inner angles of the triangle using a Law of Cosines

a**2 = b**2+c**2 - 2bc cos alpha ; ; alpha = arccos( fraction{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 5.4**2+2.34**2-3.3**2 }{ 2 * 5.4 * 2.34 } ) = 20° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 3.3**2+2.34**2-5.4**2 }{ 2 * 3.3 * 2.34 } ) = 145° 58'1" ; ; gamma = 180° - alpha - beta = 180° - 20° - 145° 58'1" = 14° 1'59" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 2.16 }{ 5.52 } = 0.39 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 3.3 }{ 2 * sin 20° } = 4.82 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 5.4**2+2 * 2.34**2 - 3.3**2 } }{ 2 } = 3.82 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 2.34**2+2 * 3.3**2 - 5.4**2 } }{ 2 } = 0.944 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 5.4**2+2 * 3.3**2 - 2.34**2 } }{ 2 } = 4.319 ; ;





#2 Obtuse scalene triangle.

Sides: a = 3.3   b = 5.4   c = 7.80991030269

Area: T = 7.21113504486
Perimeter: p = 16.50991030269
Semiperimeter: s = 8.25545515135

Angle ∠ A = α = 20° = 0.34990658504 rad
Angle ∠ B = β = 34.03329324787° = 34°1'59″ = 0.59439867259 rad
Angle ∠ C = γ = 125.9677067521° = 125°58'1″ = 2.19985400773 rad

Height: ha = 4.37105154234
Height: hb = 2.67108705365
Height: hc = 1.8476908774

Median: ma = 6.50875759729
Median: mb = 5.35222000189
Median: mc = 2.18662016098

Inradius: r = 0.87436211091
Circumradius: R = 4.82442772603

Vertex coordinates: A[7.80991030269; 0] B[0; 0] C[2.73547628747; 1.8476908774]
Centroid: CG[3.51546219672; 0.6165636258]
Coordinates of the circumscribed circle: U[3.90545515135; -2.83333952359]
Coordinates of the inscribed circle: I[2.85545515135; 0.87436211091]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 160° = 0.34990658504 rad
∠ B' = β' = 145.9677067521° = 145°58'1″ = 0.59439867259 rad
∠ C' = γ' = 54.03329324787° = 54°1'59″ = 2.19985400773 rad

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How did we calculate this triangle?

1. Input data entered: side a, b and angle α.

a = 3.3 ; ; b = 5.4 ; ; alpha = 20° ; ; : Nr. 1

2. From angle α, side b and side a we calculate side c - by using the law of cosines and quadratic equation:

a**2 = b**2 + c**2 - 2b c cos alpha ; ; ; ; 3.3**2 = 5.4**2 + c**2 - 2 * 5.4 * c * cos 20° ; ; ; ; ; ; c**2 -10.149c +18.27 =0 ; ; p=1; q=-10.149; r=18.27 ; ; D = q**2 - 4pr = 10.149**2 - 4 * 1 * 18.27 = 29.9157119227 ; ; D>0 ; ; ; ; c_{1,2} = fraction{ -q ± sqrt{ D } }{ 2p } = fraction{ 10.15 ± sqrt{ 29.92 } }{ 2 } ; ; c_{1,2} = 5.07434015 ± 2.73476287467 ; ; c_{1} = 7.80910302467 ; ; c_{2} = 2.33957727533 ; ; ; ; : Nr. 1
 text{ Factored form: } ; ; (c -7.80910302467) (c -2.33957727533) = 0 ; ; ; ; c > 0 ; ; ; ; c = 7.809 ; ; : Nr. 1
Now we know the lengths of all three sides of the triangle and the triangle is uniquely determined. Next we calculate another its characteristics - same procedure as calculation of the triangle from the known three sides SSS.

a = 3.3 ; ; b = 5.4 ; ; c = 7.81 ; ;

3. The triangle circumference is the sum of the lengths of its three sides

p = a+b+c = 3.3+5.4+7.81 = 16.51 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 16.51 }{ 2 } = 8.25 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 8.25 * (8.25-3.3)(8.25-5.4)(8.25-7.81) } ; ; T = sqrt{ 52 } = 7.21 ; ;

6. Calculate the heights of the triangle from its area.

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 7.21 }{ 3.3 } = 4.37 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 7.21 }{ 5.4 } = 2.67 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 7.21 }{ 7.81 } = 1.85 ; ;

7. Calculation of the inner angles of the triangle using a Law of Cosines

a**2 = b**2+c**2 - 2bc cos alpha ; ; alpha = arccos( fraction{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 5.4**2+7.81**2-3.3**2 }{ 2 * 5.4 * 7.81 } ) = 20° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 3.3**2+7.81**2-5.4**2 }{ 2 * 3.3 * 7.81 } ) = 34° 1'59" ; ; gamma = 180° - alpha - beta = 180° - 20° - 34° 1'59" = 125° 58'1" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 7.21 }{ 8.25 } = 0.87 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 3.3 }{ 2 * sin 20° } = 4.82 ; ; : Nr. 1

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 5.4**2+2 * 7.81**2 - 3.3**2 } }{ 2 } = 6.508 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 7.81**2+2 * 3.3**2 - 5.4**2 } }{ 2 } = 5.352 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 5.4**2+2 * 3.3**2 - 7.81**2 } }{ 2 } = 2.186 ; ;
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